This work is distributed as a Discussion Paper by the

STANFORD INSTITUTE FOR ECONOMIC POLICY RESEARCH

SIEPR Discussion Paper No. 00-25

A Purchasing PowerParity Paradox

Asaf ZussmanStanford University

January 2001

Stanford Institute for Economic Policy ResearchStanford University

Stanford, CA 94305(650) 725-1874

The Stanford Institute for Economic Policy Research at Stanford University supports research bearing oneconomic and public policy issues. The SIEPR Discussion Paper Series reports on research and policyanalysis conducted by researchers affiliated with the Institute. Working papers in this series reflect the viewsof the authors and not necessarily those of the Stanford Institute for Economic Policy Research or StanfordUniversity.

1

A Purchasing Power Parity Paradox*

Asaf Zussman

[January 2001]

Abstract

The Balassa-Samuelson hypothesis is considered the most important structural model oflong-run deviations from purchasing power parity (PPP). I present a simple model thatbuilds on the Balassa-Samuelson hypothesis and in which, paradoxically, PPP necessarilyholds in the long run. In this model real exchange rates converge because productivitylevels converge. The model’s predictions are tested empirically for 24 OECD countriesin the period 1950-92. Overall, the analysis supports the prediction that productivityconvergence is accompanied by real exchange rate convergence.

* I would like to thank William Baumol, Jeffrey Bergstrand, Menzie Chinn, CharlesEngel, Charles Jones, Michael Kumhof, Anne Krueger, Peyron Law, Ronald McKinnon,Kanda Naknoi, Assaf Razin and participants of the international economics “brown bag”seminar at Stanford University for their helpful suggestions and comments on previousversions of this paper.

2

1 Introduction

This paper attempts to marry the purchasing power parity (PPP) and growth (in

particular “convergence club”) literatures. It raises and tests a very simple hypothesis:

that convergence in levels of per-capita income within a group of countries implies

convergence towards absolute PPP between them. The motivation for this exercise is

mainly empirical and it relies on two stylized facts. The first is that at any given point in

time rich countries tend to have higher same currency prices relative to poor ones. The

second stylized fact – the existence of convergence clubs - is that within specific groups

of countries and during specific periods of time, an inverse relationship can be found

between a country’s initial level of per-capita income and its rate of growth of per capita

income over time. Combining these two stylized facts, we can expect to find a third one:

within a convergence club there should be convergence towards absolute PPP.

At the heart of the paper’s analysis is the venerable Balassa-Samuelson (BS)

hypothesis. The BS hypothesis, advanced almost 40 years ago, is considered the most

important structural model of long-run deviations from PPP. The novelty of this paper’s

approach is that it shows that ultimately the BS hypothesis does not necessarily imply

that PPP does not hold in the long run. I present a model that augments the BS

framework with a technological diffusion mechanism to yield convergence in both per

capita incomes and in real exchange rates across countries over time.

Section 2 positions the paper within the existing literature and motivates the

theoretical and empirical analysis. Section 3 presents the theoretical model. Empirical

testing of the model’s predictions is carried out in section 4. Section 5 concludes.

3

2 The existing literature

PPP and structural determinants of the real exchange rate

PPP is a simple empirical proposition that, once converted into a common

currency, national price levels should be equal. The building block of PPP is the law of

one price, which says that goods market arbitrage will equate prices of traded goods

across countries. If all goods are traded and if weights of all goods in aggregate price

indices are identical across countries, aggregate price levels will be equated. Thus

according to PPP, the real exchange rate, which can be defined as the ratio of two

countries’ price levels, expressed in a common currency, should be equal to unity, for all

pairs of countries and at all times.

Since data on absolute price levels across countries is hard to obtain, empirical

testing has concentrated on relative PPP, which says that the real exchange rate should be

constant, or at least stationary, over time. In recent years there is a growing consensus in

the empirical literature that (relative) PPP tends to hold in the long run, i.e. that real

exchange rates are stationary with a very slow rate of mean reversion. In many cases

half-lives of deviations from relative PPP are estimated to be in the range of 3 to 5 years.1

On the other hand a well established stylized fact is that when all countries’ price

levels are translated to a common currency at prevailing nominal exchange rates, rich

countries tend to have higher price levels than poor ones.2 This inverse relationship

between a country’s relative per-capita income level and its real exchange rate obviously

1 The empirical PPP literature is very extensive. Some examples include Cheung and Lai (2000), Diebold

et. al. (1991), Edison et. al. (1997), Engel (1999, 2000), Engel et. al. 1996, Engel and Kim (1999),Frankel and Rose (1996), Jorion and Sweeney (1996), Koedijk et. al. (1998), Lothian (1997), Lothian(1998), Lothian and Taylor (1996), Oh (1996), Panos et. al. (1997), Papell (1997), Papell and Theodoridis(1998), Rogers and Jenkins (1995), Taylor (2001a, 2001b). It has to be emphasized that the consensus on

4

contradicts absolute PPP. A few theoretical explanations for this deviation from absolute

PPP have been offered, most of them focus on the distinction between tradable and non-

tradable goods.3 The most prominent among these explanations is the BS hypothesis.4 A

generalized formal model that captures the BS argument is presented in section 3. Here I

will concentrate on its basic features. Balassa and Samuelson argued that productivity

tends to be higher in rich countries than in poor ones and that this tendency is more

pronounced in the tradable than in the non-tradable goods sector. Assuming perfect

international capital mobility, wage equalization across sectors in each country and the

law of one price for tradable goods,5 this pattern of productivity differentials between

sectors and across countries will lead to higher non-tradable goods prices in rich

countries. Assuming identical shares of tradable and non-tradable goods in the overall

price indices across countries, this means that rich countries will tend to have higher price

levels and appreciated real exchange rates relative to poor ones. Moreover, even if the

productivity differential between two countries is identical in both sectors, the real

exchange rate in the rich one will still be appreciated if non-tradables are more labor

intensive than tradables. This is the Baumol-Bowen hypothesis that is also formally

analyzed in section 3.

PPP is not complete. For recent challenges see Engel (2000). For potential biases in the empiricalliterature see Taylor (2001b).

2 See for example Heston et. al. (1994), Rogoff (1996) and Summers and Heston (1991).3 See Bergsrand (1991).4 The original references are Balassa (1964) and Samuelson (1964). Discussions of the model can be

found in Asea and Corden (1994), Dornbusch (1987), Froot and Rogoff (1995), Rogoff (1996).5 The validity of the last assumption is contested in the empirical literature. Barriers to trade (tariff and

non-tariff), transportation costs, imperfect competition and other factors drive a wedge between prices oftraded goods in different countries. The first major attack on the empirical validity of the Law of OnePrice is Isard (1977). The literature on this issue was surveyed recently by Goldberg and Knetter (1997).The other assumptions underlying the BS argument could also, of course, be subject to criticism.

5

A second, related, theory that also predicts that rich countries will have

appreciated real exchange rates is due to Bhagwati (1984) and others. It focuses on

differences in endowments of capital and labor across countries. Rich countries tend to

have higher capital-labor ratios than poor countries (because of imperfect capital

mobility) and thus higher marginal product of labor and higher wages.6 Assuming again

that non-tradables (largely services) are labor-intensive relative to tradables (largely

commodities), non-tradables will tend to be cheaper in poor countries than in rich ones.

A third explanation is demand oriented.7 It suggests that, assuming non-

homothetic tastes, price levels are higher in countries with higher per capita income

because non-tradables are luxuries in consumption while tradables are necessities.

Countries with higher real per-capita income will exhibit, in equilibrium, stronger

demand for non-tradables relative to tradables, raising their relative price.8

The cross-section evidence on the relationship between relative incomes and real

exchange rates tends to raise the question: does it have a time-series counterpart? The

literature answers this question only indirectly, mainly by empirically testing variants of

the BS hypothesis. In the time-series context the BS proposition is that the real exchange

rate of country i relative to country j will tend to appreciate when i’s rate of growth of

productivity relative to j’s is faster in the tradable than in the non-tradable sector. The

results coming from this line of research lend some support to the BS hypothesis:

6 This argument assumes that factor endowment differences between rich and poor countries are

sufficiently great that factor-price equalization does not hold.7 See Bergstrand (1991, 1992).8 Of course, factors other than national income (including its supply and demand elements) influence real

exchange rates across countries. Some of the variables suggested in the literature as determinants of thereal exchange rate are the share of government expenditures (and specifically military expenditures) inGDP, terms of trade, foreign aid, relative abundance in mineral resources and others. See: Bergsrand(1992), Clague (1986), Connolly and Devreux (1992), Devreux and Connolly (1996), Edwards and van

6

differential rates of growth of productivity across sectors and countries do, in certain

cases, influence real exchange rates behavior.9

The findings of the two lines of research presented above are to some extent

contradictory. Those of the “structural” literature say that we should expect real

exchange rates to follow productivity growth differentials and other real, structural,

factors. Thus long run trends of appreciation or depreciation of the real exchange rate are

possible. On the other hand the findings of the “non-structural” PPP literature, which

uses price and nominal exchange rate data only, say that we should not expect any long-

run trend in the real exchange rates. A natural environment for examining the validity of

the competing claims of the two schools is one in which different countries have very

different growth experiences over time. In such an environment one might expect real

exchange rates to have changed most dramatically over time. One such environment is a

convergence club.

Convergence clubs

Two main concepts of convergence appear in the growth literature. They are

termed β-convergence and σ-convergence. We say that there is absolute β-convergence

if poor economies tend to grow faster than rich ones. The concept of σ-convergence can

be defined as follows: a group of economies are converging in the sense of σ if the

dispersion of their real per-capita income levels tends to decrease over time. The two

Wijnbergen (1987), Kravis and Lipsey (1988), Neary (1988). Nonetheless, it is still the case that relativeper-capita national income is the best single explanatory variable of real exchange rates across countries.

9 There is a large body of empirical literature analyzing the behavior of real exchange rate over time usingthe differential productivity approach. In many cases this variable is augmented by other explanatoryvariables such as government expenditures and the terms of trade. See for example Amano and vanNorden (1995), Bahmani-Oskooee and Rhee (1996), Canzoneri et. al. (1999), Chinn (1997, 2000), Chinnand Johnston (1997), De Gregorio, Giovannini and Krueger (1994), De Gregorio, Giovannini and Wolf(1994), De Gregorio and Wolf (1994), Dutton and Strauss (1997), Edison and Klovland (1987), Engel

7

concepts are, of course, related: β-convergence is a necessary, but not a sufficient

condition for σ-convergence.10 Real exchange rate levels are heavily influenced by

monetary factors, especially in the short and the medium runs, and thus are prone to be

much more volatile than per-capita income levels. Because of this fact it would be much

harder for us to detect σ-convergence in our data. Therefore the focus of this paper will

be on β-convergence.

One of the key predictions of the neoclassical growth model is that the growth

rate of an economy will be positively related to the distance that separates it from its own

steady state. This is the concept known in the growth literature as conditional β-

convergence. To facilitate the distinction, the concept of β-convergence discussed above

is sometimes called absolute convergence. The conditional and the absolute convergence

hypotheses coincide only if all the economies have the same steady state. In order to test

the hypothesis of unconditional convergence we need a group of countries that are

expected to have the same steady state. A set of such countries would then tend to form a

“convergence club": a group of countries in which initially poor economies tend to grow

faster than the rich ones and thus catch-up or converge.

By testing for a negative relationship between average annual rates of growth and

initial levels of income, Baumol (1986) concluded that industrial countries appear to

belong to one convergence club, middle income countries to a separate, only moderately

converging club, and that low income countries actually diverged over time.11 Following

this analysis a large body of literature has tackled the empirical question of growth and

and Kim (1999), Ito (1997), Hsieh (1982), Marston (1987, 1990), Micossi and Milesi Ferretti (1994) andStrauss (1995, 1996, 1999) among others

10 For discussion of these concepts see Galor (1996) and Sala-i-Martin (1996).11 A similar early analysis of convergence is Abramovitz (1986).

8

convergence. The consensus that emerged in this literature suggested an absence of

global convergence to a single steady state. Yet the existence of a convergence club

among the top end of the income spectrum, specifically among OECD countries, is a

stylized fact of the empirical growth literature.12

12 See Barro and Sala-i-Martin (1992, 1995), and Sala-i-Martin (1996). For convergence among OECD

countries see Dowric and Nguyen (1989).

9

3 Model

This section presents a simple theoretical model that consists of three basic

elements: (1) the BS model in its cross-section dimension; (2) the BS model in its time-

series dimension; (3) a simple technological diffusion mechanism. The combination of

these three elements yields the result of simultaneous dual convergence: in levels of per

capita income and in real exchange rates. In other words under the assumptions of the

model absolute PPP necessarily holds in the long run.

I will assume a small open economy that takes the world real rate of interest as

given. Two goods are produced in the economy, traded and non-traded, according to the

following constant returns to scale Cobb-Douglas production functions:

1( ) ( )T T T Tt t t tY L Kα αθ −= (3.1)

1( ) ( )N N N Nt t t tY L Kβ βθ −= (3.2)

where , , ,i i i it t t tY L Kθ are output, productivity (or technology) level, labor input and capital

input at time t in sector i (i being T or N), while α and β are production function

parameters. Taking first order conditions with respect to capital and labor for each sector

yields the following equations:

(1 ) ( )T Tt t tR k αα θ −= − (3.3)

1( )T Tt t tW k ααθ −= (3.4)

(1 ) ( )N Nt t t tR P k ββ θ −= − (3.5)

1( )N Nt t t tW P k ββθ −= (3.6)

where tR and tW are the world real rate of interest and domestic wage rate, respectively,

itk is capital per worker in sector i and tP is the relative price of non-traded goods (i.e.

10

N Tt t tP P P= ). Solving for the relative price of non-traded goods we get the following

expression:

( ) ( ) ( ) ( )1

1 11 1

1

(1 )

(1 )

N T N Tt t t t t t tP R R

βα

β βα β βα αα α

ββ

α αθ θ γ θ θ

β β

− − −− − −

− = ≡

−

(3.7)

Note that if α=β then Tt

t Nt

Pθθ

= .

I will now define national price level in country i and time t, ,i tNPL , as a weighted

average of traded and non-traded goods prices. For simplicity I will assume that the

weights are constant and equal across time and countries.13 Therefore for every i and t

we have:

( ) ( )1

, , ,N T

i t i t i tNPL P Pδ δ−

= (3.8)

where ,N

i tP and ,T

i tP are the prices of the non-traded and traded goods, respectively. The

real exchange rate of country i relative to country j at time t is then defined as follows:

1

, , ,, ,

, , ,

N Tj t j t j t

i j t N Ti t i t i t

NPL P PQ

NPL P P

δ δ−

= = (3.9)

Assuming that purchasing power parity holds for traded goods, i.e. , ,T T

i t j tP P= ,

recalling the definition of the relative price of non-traded goods and using (3.7) we get:

( )( )

, ,

, ,

, ,

T Tj t i t

i j t N Nj t i t

Q

δβαθ θ

θ θ

=

(3.10)

13 This is the usual assumption made in the literature and is based on a Cobb-Douglas utility function with a

unit elasticity of substitution.

11

If j is a richer country than i, we would expect productivity levels in j to be higher

than in i. Moreover, we would expect the productivity ratio to be higher in the tradable

sector. According to the BS hypothesis this difference in productivity ratios is what

causes the real exchange rate of i to be depreciated relative to j. Formally: if

, , , ,T T N Nj t i t j t i tθ θ θ θ> and α=β then , , 1i j tQ > . On the other hand, if , , , , 1T T N N

j t i t j t i tθ θ θ θ= > and

α<β then we still get , , 1i j tQ > , as Baumol and Bowen have argued.

So far the discussion has concentrated on comparing price level across countries

at a given point in time. To see the time-series counterparts of these results we take time

derivatives of (3.10) to yield:

, , , ,, ,, , , ,

, , , , , ,

T NT Ni j t j t j ti t i t N T

i j t i j tT T N Ni j t j t i t j t i t

Qa a

Q

θ θθ θβ βδ δ

α θ θ θ θ α

= − − − = −

& & && &(3.11)

where , ,Ti j ta and , ,

Ni j ta are the differential productivity growth rates between countries i

and j in the tradable and the non-tradable sectors, respectively. Equation (3.11) says that,

assuming α=β, if the differential productivity growth is higher in the tradable sector than

in the non-tradable sector, country i will experience a real appreciation over time relative

to country j. If α<β, then even a balanced differential productivity growth will lead to

real appreciation.

The novel element in our theoretical framework comes from augmenting it by an

exogenous technological diffusion mechanism. 14 I will assume the following equations

for the rate of growth of productivity in the two sectors:

14 A similar specification of exogenous technological diffusion mechanism can be found in Dowrick and

Nguyen (1989) and Bernard and Jones (1996a, 1996b).

12

, ,

, ,

1T Ti t L tT

T Ti t i t

θ θφ

θ θ

= + −

&(3.12)

, ,

, ,

1N Ni t L tN

N Ni t i t

θ θφ

θ θ

= + −

&(3.13)

where Tφ and Nφ are positive productivity growth parameters, while ,TL tθ and ,

NL tθ are the

time t productivity levels in the economy with the highest productivity level (I will

assume that the same country has the productivity leadership in the two sectors).

Plugging (3.12) and (3.13) into (3.11) yields:

, , , ,

, , , ,

1 1T N

i L t L t L t

T Ni L t i t i t

Q

Q

θ θβδ

α θ θ

= − − −

&(3.14)

Thus if α=β country i will experience real appreciation relative to the productivity leader

as long as it is closer (in ratio terms) to the leader in non-tradable sector productivity than

in tradable sector productivity. When α<β country i will experience real appreciation

even if the productivity ratio between itself and the leader is the same across the two

sectors. If we make the simplifying assumption that non-tradable productivity is constant

and equal across time and countries, then real appreciation will result from tradable sector

productivity catch-up.

The model presented above implies that over time countries are catching up to the

leader, not only in terms of productivity, but also in terms of wages, capital-labor ratios,

per capita incomes and real exchange rates. Dual convergence is thus a result of

technological catch-up.

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4 Empirical analysis

Does the data exhibit the pattern of dual convergence predicted by the model? In

this section I will attempt to show that to a large extent the answer to this question is

positive. Within the group of OECD countries almost all had in 1950 a low level of per-

capita GDP and a depreciated real exchange rate relative to the U.S.; subsequently the

vast majority of these countries grew faster than the U.S. and experienced real exchange

rate appreciation. Thus convergence in per-capita GDP levels was accompanied by

convergence towards absolute PPP.

The database used in the empirical analysis is the Penn World Tables (PWT).

The PWT displays a set of national accounts economic time series. Its expenditure

entries are denominated in a common set of prices in a common currency so that real

quantity comparisons can be made, both between countries and over time. In its latest

version, which will be used in the paper, there are 152 countries and 43 years (1950-

1992). The analysis here focuses on 24 OECD countries.15

In this paper the variable relative per-capita GDP, Y, is defined as real GDP per-

capita, in current international prices, relative to the U.S. The variable y is the natural

logarithm of Y. The real exchange rate, Q, is defined as 100/P where P is the price level

of GDP (%) [(PPP of GDP relative to the U.S. dollar)/(exchange rate relative to the U.S.

dollar)]. The variable q is the natural logarithm of Q. For extended discussion of the

variables and the data set see Heston and Summers (1991). Figure 1 is a scatter diagram

15 The countries are Australia, Austria, Belgium, Canada, Denmark, Finland, France, West Germany,

Greece, Iceland, Ireland, Italy, Japan, Luxembourg, The Netherlands, New Zealand, Norway, Portugal,Spain, Sweden, Switzerland, Turkey, United Kingdom and the United States. These include the twentyoriginal (1961) members plus four members that joined the OECD by 1973. Current OECD membersthat joined only recently, namely the Czech republic, Hungary, Korea, Mexico and Poland, are excludedfrom the analysis.

14

of all (986) observations on relative per-capita GDP (Y) and real exchange rate (Q) for

the 23 countries during 1950-92.16 Visual inspection seems to indicate a strong negative

relationship between relative per-capita GDP and real exchange rate.

Figure 2 plots the time series of relative per-capita GDP and real exchange rate

for the twenty-three countries. For the majority of the countries a negative relationship

between the two variables is again apparent. A striking feature of the graphs displayed in

Figure 2 is that during the 1980s most countries experienced first a substantial real

depreciation and then a large real appreciation relative to the U.S. Dollar. This fact

points to the important influence of other variables, in this case mainly monetary, on the

behavior of real exchange rates.

The next stage in our analysis is more formal. We will perform a procedure to

test for unit roots in the individual time series, assuming that the data generating process

is unknown. The procedure, which is outlined in Enders (1995), has four steps. In the

first step we estimate the least restrictive model, which contains both a constant and a

time trend: 0 1 2 1

2

t t i t i t

p

i

y a y a t yγ β ε− − +

=

∆ = + + + ∆ +∑ . We then use the ττ statistic to test the

null hypothesis γ=0. If the null hypothesis of a unit root is rejected we conclude that the

series does not contain a unit root. Otherwise we go to step 2. In this step we have to

determine whether too many deterministic regressors were included in Step 1. We thus

test for the significance of the trend term under the null of a unit room (i.e. we use the

τβτ statistic to test the significance of a2). If the trend is not significant we proceed to

Step 3. Otherwise we retest for the presence of a unit root using the standardized normal

16 Greece has data only for the period 1950-91; Portugal for the period 1950-90.

15

distribution (z). If the null of a unit root is rejected we conclude that the series does not

contain a unit root. Otherwise we conclude that it does.

In Step 3 we run the regression without the trend. We test for the presence of a

unit root using the τµ statistic. If the null is rejected we conclude that the series does not

contain a unit root. Otherwise we test for the significance of the constant using the

ταµ statistic. If the drift is not significant proceed to Step 4. Otherwise we test for the

presence of a unit root using the standardized normal distribution. If the null hypothesis

of a unit root is rejected, we conclude that the series does not contain a unit root.

Otherwise we conclude that it does. In Step 4 we run the regression without the trend and

the constant. We use the τ statistic to test for the presence of a unit root. If the null

hypothesis of a unit root is rejected we conclude that the series does not contain a unit

root. Otherwise we conclude that it does. We first apply the procedure to the real

exchange rate series (q). Results are presented in Table 1.

Table 1: Unit-root tests for the real exchange rate (q)

Step 1 Step 2 Step 3 Step 4

Country τ−ττ−τ LL τ−βττ−βτ z τ−µτ−µ LL τ−αµτ−αµ ττ LL Conclusion

NEW ZEALAND -4.4321*** 1 Stationary

U.K. -4.3845*** 2 Stationary

IRELAND -4.3105*** 2 Stationary

GREECE -3.6051** 2 Stationary

FINLAND -3.5892** 4 Stationary

JAPAN -3.5608** 1 Stationary

TURKEY -3.2386* 1 Stationary

FRANCE -3.2234* 2 Stationary

SWEDEN -3.1929* 1 Stationary

AUSTRIA -3.1292 1 -3.0851** *** Stationary

ITALY -2.9839 1 -2.9585** *** Stationary

SWITZERLAND -3.0880 4 -2.9584** *** Stationary

SPAIN -3.0656 1 -2.7892* *** Stationary

NORWAY -3.0143 2 -2.7555* *** Stationary

LUXEMBOURG -3.1078 1 -2.7438* *** Stationary

DENMARK -2.9860 4 -2.6973* *** Stationary

W. GERMANY -2.9424 1 -2.6171* *** Stationary

16

ICELAND -3.1690 1 0.4000 -3.2124** 2 Stationary

PORTUGAL -3.0740 1 -1.1315 -2.8962* 1 Stationary

BELGIUM -3.1536 1 -2.1544 -2.2255 1 0.4874 -2.2501** 1 Stationary

CANADA -3.0316 1 1.9360 -2.2997 1 -0.9139 -2.1535** 2 Stationary

AUSTRALIA -2.1844 1 -1.1020 -0.8302 1 0.9310 -1.8966* 1 Stationary

NETHERLANDS -2.6186 4 -2.2517 -1.2677 4 -0.1333 -1.7186* 3 Stationary

Source: Penn World Tables and author’s calculations. Notes: *, **, *** denote, respectively, significance at the 10%, 5%, 1% levels;LL denotes lag-length. We allow for up to 4 lags. Critical values (for T=50) for the tests are given in Dickey and Fuller (1981) andHamilton (1994).

As Table 1 shows, application of the procedure to the real exchange rate (q) series

leads to rejection of the null hypothesis and to the conclusion that in all cases the series

does not contain a unit root. We next apply the procedure to the relative per-capita GDP

(y) series. Results are presented in Table 2:

Table 2: Unit-root tests for relative per-capita GDP (y)

Step 1 Step 2 Step 3 Step 4

Country τ−ττ−τ LL τ−βττ−βτ z τ−µτ−µ LL τ−αµτ−αµ ττ LL Conclusion

AUSTRALIA -5.7035*** 1 Stationary

AUSTRIA -4.1341** 3 Stationary

W. GERMANY -4.1203** 3 Stationary

TURKEY -4.0487** 0 Stationary

IRELAND -3.8857** 3 Stationary

NEW ZEALAND -3.6550** 3 Stationary

LUXEMBOURG -3.3227* 4 Stationary

PORTUGAL -3.0881 3 3.1383*** *** Stationary

ICELAND -2.9098 1 2.3914* *** Stationary

SWEDEN -2.3782 3 -0.3058 -3.6724*** 3 Stationary

SWITZERLAND -3.0708 3 0.3961 -3.4365** 3 Stationary

NETHERLANDS -2.1996 1 1.0270 -3.1880** 1 Stationary

FRANCE -1.7222 3 0.3859 -3.1847** 3 Stationary

GREECE -0.6919 3 -0.1274 -3.0158** 2 Stationary

JAPAN -1.5325 3 0.8810 -2.8249* 4 Stationary

ITALY -2.3157 1 1.6258 -2.7845* 3 Stationary

SPAIN -2.2982 3 1.2162 -2.6248* 3 Stationary

DENMARK -2.1189 1 1.1095 -2.2844* 2 -1.4791 -2.7521*** 2 Stationary

BELGIUM -2.4094 4 2.2897 -1.0967 3 -0.3732 -2.6218*** 1 Stationary

FINLAND -2.6067 3 2.2092 -2.2965 3 -1.4317 -2.5582** 4 Stationary

NORWAY -2.1077 4 1.8763 -1.3889 1 -0.7908 -2.2111** 0 Stationary

U.K. -2.5678 0 1.8063 -2.0299 1 -1.8242 -2.0775** 2 Stationary

CANADA -2.2826 2 2.2072 -1.0677 4 -0.2561 -1.7745* 3 Stationary

17

Source: Penn World Tables and author’s calculations. Notes: *, **, *** denote, respectively, significance at the 10%, 5%, 1% levels;LL denotes lag-length. We allow for up to 4 lags. Critical values (for T=50) for the tests are given in Dickey and Fuller (1981) andHamilton (1994).

As Table 2 shows, application of the procedure to the relative per-capita GDP

series leads to rejection of the null and to the conclusion that in all cases this series too

does not contain a unit root.

So far the formal analysis concentrated on the statistical properties of the series.

The more interesting questions, of course, concern the economics behind them. The first

issue that we have to address is the speed of reversion, which is captured in the parameter

γ. When the regression includes a constant only this parameter captures the speed of

mean reversion. In the context of the real exchange rate literature it measures the speed

of reversion to relative PPP. When the regression includes both a constant and a time

trend this parameter captures the speed of reversion to a trend. Table 3 contains data on

the average reversion parameter in the unit-root tests performed above for the twenty-

three individual time series.

Table 3: The reversion parameter, γγ

Real exchange rate – q Relative GDP per capita – yWith constant and

time trendWith constant

onlyWith constant and

time trendWith constant

onlyγ IHL γ IHL γ IHL γ IHL

Average -0.3369 1.6871 -0.1434 4.4783 -0.3239 1.7706 -0.1454 4.4122

Source: Penn World Tables and author’s calculations. IHL stands for “implied half

life”.

The results presented in the table are quite interesting. First we note the similarity

in the magnitude of the average γ coefficient for the regressions with a constant and a

trend: -0.3669 for the real exchange rate series and –0.3239 for the relative per-capita

GDP series. Second we note the similarity in the magnitude of the average γ for the

18

regressions with only a constant: -0.1434 for the real exchange rate series and –0.1454 for

the relative per-capita GDP series.

The most important observation in Table 3 is this: there is a large difference

between the average γ coefficients for the real exchange rate series between the

regressions with a constant and a time trend and those with only a constant.17 As

mentioned earlier, in essence the regression with only a constant tries to capture reversion

to relative PPP. The half-life of deviation from relative PPP implied by the value of the γ

coefficient is 4.48 years. This figure fits well with the “consensus” estimate of half-lives

of deviations from relative PPP discussed in Rogoff (1996): three to five years.18 The

fact that deviations are so persistent led Rogoff (1996) to coin the term “The Purchasing

Power Parity Puzzle”. But as shown above, when one includes a trend in the regressions

the implied half-life drops to only 1.69 years. A similar result was obtained by Taylor

(2001a) for a group of twenty countries in the post World War II years. The question

then is this: what explains the trend in the real exchange rate? The answer proposed here

is, of course, the process of dual convergence.

Table 4 provides evidence on the behavior on relative per-capita GDP and the real

exchange rate over time for all twenty-three countries. The column labeled Y0 shows the

initial, 1950, relative per-capita GDP. The next column shows the trend-change in the

natural logarithm of relative per-capita GDP, y, which was obtained by an OLS

regression of the natural logarithm of relative per-capita GDP on a constant and a linear

17 There is, of course, a corresponding difference in the relative per-capita GDP series figures, but here we

concentrate on the real exchange rate series.18 See also Obstfeld and Rogoff (2001)

19

time trend.19 The next two columns show the initial real exchange rate (Q0) and the

trend-change in the natural logarithm of the real exchange rate, q. The last two columns

attempt to estimate the long-run relationship between y and q. The penultimate column

shows the slope coefficient from an OLS regression of y on a constant and q.20 The last

column shows the intercept coefficient from that regression.

Table 4: Relative per-capita GDP and real exchange rate behavior

Regression of y on q and cCountry Y0 Trend in y Q0 Trend in q

Slope Intercept

Australia 0.8332 0.0019*** 1.5657 -0.0107*** -2.3911*** -0.3953***Austria 0.3251 0.0177*** 1.6450 -0.0192*** -0.9012*** -0.3546***Belgium 0.5071 0.0102*** 1.2718 -0.0098*** -1.0609*** -0.4086***Canada 0.7058 0.0090*** 1.0346 0.0026*** 0.2533*** 0.0134

Denmark 0.5915 0.0079*** 1.5610 -0.0212*** -1.8969*** -0.6040***Finland 0.3875 0.0147*** 1.2089 -0.0118*** -0.6994*** -0.3699***France 0.4573 0.0126*** 1.2560 -0.0071*** -0.4599*** -0.1596***

W. Germany 0.3776 0.0144*** 1.4196 -0.0165*** -0.9149*** -0.2875***Greece 0.1557 0.0242*** 0.9843 0.0035* 0.1123 0.4060***Iceland 0.3975 0.0189*** 0.7336 0.0028 0.0790 -0.2531***Ireland 0.3041 0.0130*** 1.6810 -0.0130*** -0.9489*** -0.6820***Italy 0.3152 0.0182*** 1.6431 -0.0145*** -0.7067*** -0.1897***Japan 0.1620 0.0382*** 2.2800 -0.0295*** -0.7066*** -0.3419***

Luxembourg 0.6707 0.0031*** 1.3875 -0.0126*** -0.6246* -0.0222Netherlands 0.5134 0.0092*** 1.8793 -0.0212*** -2.0049*** -0.6470***

New Zealand 0.8091 -0.0029*** 1.4255 -0.0031** 0.5403** 0.4590**Norway 0.4772 0.0130*** 1.4269 -0.0181*** -1.1460*** -0.5889***Portugal 0.1421 0.0231*** 1.6767 -0.0026 -0.1659** 0.3563**

Spain 0.2192 0.0192*** 1.8165 -0.0157*** -0.6253*** -0.1287Sweden 0.6597 0.0042*** 1.3963 -0.0148*** -1.9434*** -0.5215***

Switzerland 0.7635 0.0034*** 1.5161 -0.0261*** -1.9621*** -0.1364Turkey 0.1264 0.0076*** 1.4545 0.0163*** 0.9957*** 2.3695***U.K. 0.5889 0.0035*** 1.5635 -0.0104*** -2.2066*** -0.7195***

Average 0.4561 0.0124 1.4708 -0.0110 -0.8428 -0.1394

Source: Penn World Tables and author’s calculations. Notes: *, **, *** denote,

respectively, significance at the 10%, 5%, 1% levels in the relevant tests.

Did the OECD countries indeed form a convergence club? The data presented in

Table 4 shows that they did: all countries started out in 1950 with a lower per-capita GDP

than the U.S. and all, except for New Zealand, grew faster than the U.S. during the

following four decades. What happened to the real exchange rate of those countries

19 Natural logarithm of the original series is used in order to preserve consistency with the tests performed

above.

20

during this period? There is statistically strong evidence for the hypothesis of initial

depreciated real exchange rate and subsequent real appreciation in eighteen out of

twenty-three cases.21 The second to last column shows that in those eighteen cases there

is evidence for a long run inverse relationship between relative per-capita GDP and the

real exchange rate.22 The evidence presented in the last column lead to the conclusion

that although the real exchange rates are converging, they are not converging to unity as

implied by the model. If absolute PPP holds when relative per-capita GDP is unity, then

the intercept coefficient in our regression should be equal to zero. But in practice it is

significantly different from zero in nineteen cases. On average, the implied deviations

from absolute PPP are in the order of thirteen percent, i.e. if all countries had the same

per-capita GDP as the U.S. they would have, on average, prices that are thirteen percent

higher than those in the U.S. Nevertheless, Table 4 establishes that there a remarkable

amount of support for the dual β-convergence hypothesis.

20 The results from such regression are not spurious as Table 1 and 2 show that both variables are

stationary.21 Canada and Turkey started out with a depreciated real exchange rate but exhibited a statistically very

significant trend of real depreciation. Greece and Iceland started out with an appreciated currency butneither showed a very significant trend of either depreciation or appreciation. New Zealand, despite thefact that it had a lower per-capita GDP and a depreciated real exchange rate in 1950, showed astatistically very significant trend of growth divergence and real exchange rate convergence.

22 Statistical evidence against the dual convergence hypothesis is established in only three cases: Canada,New Zealand and Turkey.

21

5 Conclusion

The paper has shown that within a specific group of countries and over a specific

period of time, convergence in levels of per-capita GDP was accompanied to a large

extent by a convergence in real exchange rates. To the best of my knowledge this paper

is the first to raise and test this hypothesis of dual convergence. An important message

emerging form the exercise performed here is that real factors should be at the center of

real exchange rate analysis. Differences in technologies, endowments and tastes should

always be taken into account.

An interesting implication of the paper concerns the question of undervaluation

and overvaluation of currencies. The results presented above suggest that in order to

properly analyze the question of real undervaluation or overvaluation in the dynamic

sense we need to know the fundamental determinants of the steady state levels of per-

capita GDP of the countries involved and their current positions relative to these steady

states. This obviously makes the calculations extremely complicated.

The paper raises another interesting question. Is it possible to find a similar

relationship between convergence in levels of per-capita GDP and relative price levels

across other sets of countries and regions? Research in economic growth has shown that

a pattern of absolute convergence in levels of per-capita income characterizes earlier

historic periods (such as the late nineteenth century) and geographic and political units

(such as U.S. states). It would be interesting to examine whether in those cases, although

possibly somewhat different from the theoretical perspective, convergence in relative

prices can be detected.

22

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25

0

1

2

3

4

0.0 0.2 0.4 0.6 0.8 1.0

relative GDP per capita (US=1)

real

exc

hang

e ra

te (U

S=1

)Figure 1: Relative GDP per capita and real exchange rate

OECD, 1950-92

26

0.8

1.0

1.2

1.4

1.6

0.68

0.72

0.76

0.80

0.84

0.88

50 55 60 65 70 75 80 85 90

Australia

0.6

0.8

1.0

1.2

1.4

1.6

1.8

0.3

0.4

0.5

0.6

0.7

0.8

50 55 60 65 70 75 80 85 90

Austria

0.6

0.8

1.0

1.2

1.4

0.45

0.50

0.55

0.60

0.65

0.70

0.75

0.80

50 55 60 65 70 75 80 85 90

Belgium

0.8

0.9

1.0

1.1

1.2

0.65

0.70

0.75

0.80

0.85

0.90

0.95

1.00

50 55 60 65 70 75 80 85 90

Canada

0.6

0.8

1.0

1.2

1.4

1.6

0.50

0.55

0.60

0.65

0.70

0.75

0.80

0.85

50 55 60 65 70 75 80 85 90

Denmark

0.6

0.8

1.0

1.2

1.4

0.3

0.4

0.5

0.6

0.7

0.8

50 55 60 65 70 75 80 85 90

Finland

0.7

0.8

0.9

1.0

1.1

1.2

1.3

0.4

0.5

0.6

0.7

0.8

50 55 60 65 70 75 80 85 90

France

0.6

0.8

1.0

1.2

1.4

1.6

0.3

0.4

0.5

0.6

0.7

0.8

0.9

50 55 60 65 70 75 80 85 90

Germany

0.8

1.0

1.2

1.4

1.6

1.8

2.0

0.15

0.20

0.25

0.30

0.35

0.40

50 55 60 65 70 75 80 85 90

Greece

0.4

0.6

0.8

1.0

1.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

50 55 60 65 70 75 80 85 90

Iceland

0.8

1.0

1.2

1.4

1.6

1.8

0.25

0.30

0.35

0.40

0.45

0.50

0.55

50 55 60 65 70 75 80 85 90

Ireland

0.6

0.8

1.0

1.2

1.4

1.6

1.8

0.3

0.4

0.5

0.6

0.7

0.8

50 55 60 65 70 75 80 85 90

Italy

Figure 2: Relative GDP per capita and real exchange rate in OECD countries

0.5

1.0

1.5

2.0

2.5

0.0

0.2

0.4

0.6

0.8

1.0

50 55 60 65 70 75 80 85 90

J a p a n

0.6

0.8

1.0

1.2

1.4

1.6

0.6

0.7

0.8

0.9

1.0

50 55 60 65 70 75 80 85 90

Luxembourg

0.6

0.8

1.0

1.2

1.4

1.6

1.8

2.0

0.45

0.50

0.55

0.60

0.65

0.70

0.75

0.80

50 55 60 65 70 75 80 85 90

Q_133 Y_133

Netherlands

1.0

1.2

1.4

1.6

1.8

0.65

0.70

0.75

0.80

0.85

50 55 60 65 70 75 80 85 90

New Zealand

0.6

0.8

1.0

1.2

1.4

1.6

0.4

0.5

0.6

0.7

0.8

0.9

50 55 60 65 70 75 80 85 90

Norway

1.2

1.4

1.6

1.8

2.0

2.2

2.4

2.6

0.10

0.15

0.20

0.25

0.30

0.35

0.40

0.45

50 55 60 65 70 75 80 85 90

Portugal

0.8

1.2

1.6

2.0

2.4

0.2

0.3

0.4

0.5

0.6

50 55 60 65 70 75 80 85 90

Spain

0.6

0.8

1.0

1.2

1.4

1.6

0.65

0.70

0.75

0.80

0.85

0.90

50 55 60 65 70 75 80 85 90

Sweden

0.6

0.8

1.0

1.2

1.4

1.6

1.8

0.75

0.80

0.85

0.90

0.95

1.00

50 55 60 65 70 75 80 85 90

Switzerland

0.5

1.0

1.5

2.0

2.5

3.0

3.5

0.12

0.14

0.16

0.18

0.20

0.22

50 55 60 65 70 75 80 85 90

Turkey

0.8

1.0

1.2

1.4

1.6

0.56

0.60

0.64

0.68

0.72

0.76

50 55 60 65 70 75 80 85 90

U.K.